Construction and analysis of projected deformed products

Abstract

We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and the analysis of deformed products such that specified faces (e. g., all the k-faces) are "strictly preserved" under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices, we construct a deformed n-cube, whose projection to the last d coordinates yields a neighborly cubical d-polytope. As an extension of the cubical case, we construct matrix representations of deformed products of (even) polygons (DPPs) which have a projection to d-space that retains the complete (⌊d/2}⌋-1)-skeleton. In both cases the combinatorial structure of the images under projection is determined by the neighborly simplicial polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig and Ziegler (Discrete Comput. Geom. 24:325-344, 2000) as well as of the projected deformed products of polygons announced by Ziegler (Electron. Res. Announc. Am. Math. Soc. 10:122-134, 2004), a family of 4-polytopes whose "fatness" gets arbitrarily close to 9. © 2009 Springer Science+Business Media, LLC.